5

This is called the spectral inclusion theorem. Toeplitz's 1918 paper Das algebraische Analogon zu einem Satze von Fejér already has it for finite dimensional operators as Satz 4. This was before the language of operators and Hilbert spaces introduced by von Neumann in 1927-29, see Highlights in the History of Spectral Theory by Steen, so it is phrased in ...


3

The original version of the dominated convergence, from which the monotone convergence trivially follows assuming that the limit is Lebesgue integrable, was published by Lebesgue in Leçons sur l'Intégration et la Recherche des Fonctions Primitives (1904). This is a compilation of his lectures at Collège de France over the preceeding five years. In Sopra l'...


3

It is existence and uniqueness question for ordinary differential equations. When I was a student (in 1970s) Lipschitz functions were not omnipresent in Analysis. The only context where this name appeared (in undergraduate curriculum) was existence and uniqueness theorem for ODE. And this was apparently his original motivation, as the reference in the ...


1

It seems the only firm fact about the origin of the problem is that it is discussed for the first time in print in 1969 in Rosenthal's On quasi-complemented subspaces of Banach spaces, without attribution to Banach or Mazur. The attribution to Mazur appears in Ferrando--Kakol--Lopez-Pellicer--Sliwa paper On the separable quotient problem for Banach spaces ...


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